Second Order Problem with Nonlinear Boundary Conditions

Abstract

The chapter is devoted to the impulsive nonlinear boundary value problem $$\begin{aligned} u''(t) = f(t,u(t),u'(t)) \quad \text {for a.e. }t \in [a,b] \subset \mathbb {R}, \end{aligned}$$ $$\begin{aligned} u(t_i+) = J_i(u(t_i-)),\quad u'(t_i+) = M_i(u'(t_i-)), \quad i = 1,\ldots ,p, \end{aligned}$$ $$\begin{aligned} g_1(u(a),u(b)) = 0, \quad g_2(u'(a),u'(b)) = 0, \end{aligned}$$ where \(p\in \mathbb {N}\), \(f \in \mathrm {Car}([a,b]\times \mathbb {R}^{2})\), \(g_1\), \(g_2 \in \mathbb {C}(\mathbb {R}^2)\), \(J_i\), \(M_i \in \mathbb {C}(\mathbb {R})\), \(i=1,\ldots , p\). Impulses are considered at the fixed points \(t_1,\ldots , t_p\), \(a